Abstract

A time-domain approach has been developed that is capable of evaluating the performance of an adaptive-optics system degraded by a wide variety of effects acting together or alone. In the study conducted here the normalized antenna gain (or Strehl ratio) is evaluated for a system degraded by turbulence, anisoplanatism, a finite servo bandwidth, and a combination of anisoplanatism and a finite servo bandwidth. A diameter dependence of these effects is established, illustrating that the degrading influence associated with a finite servo bandwidth is less severe than the degrading influence associated with anisoplanatism over a wide range of diameters. For the case of a small-diameter system degraded by anisoplanatism, system performance improves slightly when a finite servo bandwidth is introduced.

© 1984 Optical Society of America

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References

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  1. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  2. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wavefront,” Proc. IEEE 55, 57–67 (1967).
    [Crossref]
  3. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–392 (1977).
    [Crossref]
  4. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wavefront compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
    [Crossref]
  5. A more-detailed treatment of the analysis presented here can be found in the following unpublished work: G. A. Tyler, “Evaluation of factors that degrade adaptive optical system performance: a time domain approach,” (The Optical Sciences Company, Placentia, Calif., December1982).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 3.
  7. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  8. D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,” J. Opt. Soc. Am. 58, 961–969 (1968).
    [Crossref]

1982 (1)

1977 (1)

1976 (1)

1968 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wavefront,” Proc. IEEE 55, 57–67 (1967).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 3.

Fried, D. L.

Greenwood, D. P.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Tyler, G. A.

A more-detailed treatment of the analysis presented here can be found in the following unpublished work: G. A. Tyler, “Evaluation of factors that degrade adaptive optical system performance: a time domain approach,” (The Optical Sciences Company, Placentia, Calif., December1982).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 3.

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wavefront,” Proc. IEEE 55, 57–67 (1967).
[Crossref]

Other (3)

A more-detailed treatment of the analysis presented here can be found in the following unpublished work: G. A. Tyler, “Evaluation of factors that degrade adaptive optical system performance: a time domain approach,” (The Optical Sciences Company, Placentia, Calif., December1982).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), Chap. 3.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

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Figures (7)

Fig. 1
Fig. 1

Normalized antenna gain due to turbulence.

Fig. 2
Fig. 2

Normalized antenna gain for a system degraded by anisoplanatism. The graph contains five curves, each corresponding to a different value of ϑ/ϑ0. The five values of ϑ/ϑ0 are 0.0001, 0.3, 1, 2, and 3.

Fig. 3
Fig. 3

Normalized antenna gain for a system degraded by a finite servo bandwidth. The graph contains 10 curves, each corresponding to a different value of fG/f3dB. Starting from the top curve, the values of fG/f3dB are 0.0001, 0.1, 0.3, 0.6, 1, 2, 3, 10, 100, and 1000. As indicated, the curves corresponding to fG/f3dB equal to 0.0001 and 0.1 are almost identical.

Fig. 4
Fig. 4

Comparison of system performance in the presence of anisoplanatism or a finite servo bandwidth. When results with similar large-diameter asymptotic values are compared, the degradation that is due to a finite servo bandwidth is not so severe as that which is due to anisoplanatism at intermediate diameters. As discussed in the text, this illustrates the degrading influence associated with the retention of distorted high-frequency phase information in the anisoplanatism case. The three cases presented in the figure have antenna gains in the large-diameter asymptotic limit corresponding to exp[−(0.3)5/3], exp[−(1)5/3], and exp[−(3)5/3].

Fig. 5
Fig. 5

Performance of an adaptive-optics system degraded by anisoplanatism and a finite servo bandwidth. Each figure corresponds to a different value of fG/f3dB. As indicated in (a)–(j), the values of fG/f3dB are 0.0001, 0.1, 0.3, 0.6, 1, 2, 3, 10, 100, and 1000. Five curves are presented in each of the figures and correspond to different values of ϑ/ϑ0. The values of ϑ/ϑ0 are 0.0001, 0.3, 1, 2, and 3. As indicated in (a), when f3dB is 10,000 times fG, the results are identical with those associated with anisoplanatism in Fig. 2. When f3dB is smaller than 0.01 times fG [(i) and (j)], the performance is similar to that which would be obtained if no adaptive optics were used, as illustrated in Fig. 1.

Fig. 6
Fig. 6

Performance of an adaptive-optics system degraded by anisoplanatism and a finite servo bandwidth. The results here are identical with those of Fig. 5. They are rearranged so that for a given level of anisoplantism the effect of a finite servo bandwidth can easily be determined. Each figure corresponds to a different value of ϑ/ϑ0. As indicated in (a)–(e), these values of ϑ/ϑ0 are 0.0001, 0.3, 1, 2, and 3. Ten curves are present in each figure and correspond to different values of fG/f3dB. The values of fG/f3dB are 0.0001, 0.1, 0.3, 0.6, 1, 2, 3, 10, 100, and 1000. In (a), ϑ/ϑ0 = 0.0001, and the results are almost identical with those associated with a finite servo bandwidth alone, as presented in Fig. 3.

Fig. 7
Fig. 7

A small servo bandwidth can improve performance. This figure is actually an enlarged portion of Fig. 6(e). The result presented here illustrates the role of high-frequency phase information. The solid curve illustrates the performance of an adaptive-optics system degraded primarily by anisoplanatism. When a smaller servo bandwidth is introduced, performance is improved because removing the distorted high-frequency components results in high antenna gain for this range of diameters. When the diameter is much larger than r0, the introduction of a smaller servo bandwidth results in reduced performance, as illustrated in Fig. 6.

Equations (64)

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I ( r ) = π 4 D 2 ( λ z ) - 2 d α K ( α ) exp [ - ½ D ( α ) ] × exp ( - 2 π i λ z r · α ) .
α = r 1 - r 2 .
K ( α ) = { 2 π { cos - 1 ( α D ) - ( α / D ) [ 1 - ( α / D ) 2 ] 1 / 2 } for α D 0 for α > D .
D ( r 1 - r 2 ) = [ ϕ ( r 1 , t ) - ϕ ( r 2 , t ) ] 2 ,
G / G DL = I ( 0 ) / I DL ( 0 ) .
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - ½ D ( α ) ] .
ϕ T ( r , t ) = k d h n ( r + ϑ ˙ h t , h ) ,
ϕ T ( r , t ) = d f Φ ( r , f ) exp ( 2 π i f t ) ,
Φ ( r , f ) = d t ϕ T ( r , t ) exp ( - 2 π i f t ) .
ϕ B ( r , t ) = d f Φ ( r , f ) H ( f ) exp ( 2 π i f t ) ,
ϕ ( r , t ) = ϕ T ( r , t ) - ϕ B ( r , t ) .
ϕ ( r , t ) = d f Φ ( r , f ) [ 1 - H ( f ) ] exp ( 2 π i f t ) .
[ n ( r , h ) - n ( r , h ) ] 2 = C N 2 ( h + h 2 ) × ( r - r 2 + h - h 2 ) 1 / 3 ,
D ( α ) = 2.91 k 2 d f d t d h 1 - H ( f ) 2 exp ( - 2 π i f t ) C N 2 ( h ) × [ ½ α + ϑ ˙ h t 5 / 3 + ½ α - ϑ ˙ h t 5 / 3 - ϑ ˙ h t 5 / 3 ] .
H ( f ) = 0             ( turbulence degradation ) .
D ( α ) = 6.88 ( α / r 0 ) 5 / 3 ,
r 0 = [ 2.91 6.88 k 2 d h C N 2 ( h ) ] - 3 / 5 .
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - 3.44 ( α / r 0 ) 5 / 3 ] .
G / G DL = 1             for D / r 0 0
G / G DL = ( D / r 0 ) - 2             for D / r 0 .
H ( f ) = exp ( - 2 π i f Δ t )             ( anisoplanatism ) ,
D ( α ) = 2 S ( α , ϑ ) ,
S ( α , ϑ ) = 2.91 k 2 d h C N 2 ( h ) ( α 5 / 3 - ½ α + ϑ h 5 / 3 - ½ α - ϑ h 5 / 3 + ϑ h 5 / 3 )
ϑ = ϑ ˙ Δ t .
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - S ( α , ϑ ) ] ,
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - 6.88 ( α / r 0 ) 5 / 3 ]             for ( D / r 0 ) / ( ϑ / ϑ 0 ) 0 ,
G / G DL = exp [ - ( ϑ / ϑ 0 ) 5 / 3 ]             for ( D / r 0 ) / ( ϑ / ϑ 0 ) .
ϑ 0 = [ 2.91 k 2 d h C N 2 ( h ) h 5 / 3 ] - 3 / 5 .
H ( f ) = 1 1 + i f / f 3 dB             ( finite servo bandwidth ) .
D ( α ) = 0 d τ exp ( - τ ) S ( α , 1.186 ϑ 0 f G τ f 3 dB ) ,
f G = [ ½ Γ ( 8 / 3 ) ] 3 / 5 ϑ ˙ / ( 2 π ϑ 0 ) .
τ = 2 π t f 3 dB
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) × exp [ - ½ 0 d τ exp ( - τ ) S ( α , 1.186 ϑ 0 f G τ f 3 dB ) ] .
S ( α , 1.186 ϑ 0 f G τ f 3 dB ) = 6.88 ( α / r 0 ) 5 / 3             for ( D / r 0 ) / ( f G / f 3 dB ) 0 ,
S ( α , 1.186 ϑ 0 f G τ f 3 dB ) = ( 1.186 f G τ f 3 dB ) 5 / 3             for ( D / r 0 ) / ( f G / f 3 dB ) .
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - 3.44 ( α / r 0 ) 5 / 3 ]             for ( D / r 0 ) / ( f G / f 3 dB ) 0 ,
G / G DL = exp [ - ( f G / f 3 db ) 5 / 3 ]             for ( D / r 0 ) / ( f G / f 3 dB ) .
H ( f ) = exp ( - 2 π i f Δ t ) 1 + i f / f 3 dB ( finite servo bandwidth and anisoplanatism ) .
D ( α ) = 0 d τ exp ( - τ ) [ 2 S ( α , J + 1.186 ϑ 0 f G τ f 3 dB ) - S ( α , 1.186 ϑ 0 f G τ f 3 dB ) ] ,
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp { - 0 d τ exp ( - τ ) × [ S ( α , ϑ + 1.186 ϑ 0 f G τ f 3 dB ) - ½ S ( α , 1.186 ϑ 0 f G τ f 3 dB ) ] } .
G / G DL = ( ¼ π D 2 ) - 1 d α K ( α ) exp [ - 3.44 ( α / r 0 ) 5 / 3 ]             for D / r 0 ϑ / ϑ 0 and D / r f G / f 3 dB 0 ,
G / G DL = exp [ - ( Θ / ϑ 0 ) 5 / 3 ]             for D / r 0 ϑ / ϑ 0 and D / r f G / f 3 dB ,
Θ = { 0 d τ exp ( - τ ) [ ( ϑ + 1.186 ϑ 0 f G τ f 3 dB ) 5 / 3 - ½ ( 1.186 ϑ 0 f G τ f 3 dB ) 5 / 3 ] } 3 / 5 .
D ( r 1 - r 2 ) = ϕ ( r 1 , t ) - ϕ ( r 2 , t ) 2 ,
β = ½ ( r 1 + r 2 ) ,
α = r 1 - r 2
D ( α ) = ϕ ( β + ½ α , t ) - ϕ ( β - ½ α , t ) 2 .
D ( α ) = d f d f [ 1 - H ( f ) ] [ 1 - H * ( f ) ] exp [ 2 π i ( f - f ) t ] × { [ Φ ( β + ½ α , f ) - Φ ( β - ½ α , f ) ] × [ Φ ( β + ½ α , f ) - Φ ( β - ½ α , f ) ] } .
D ( α ) = d f d f d t 1 d t 2 [ 1 - H ( f ) ] × [ 1 - H * ( f ) ] exp [ 2 π i ( f - f ) t ] exp [ - 2 π i ( f t 1 - f t 2 ) ] × { [ ϕ T ( β + ½ α , t 1 ) - ϕ T ( β - ½ α , t 1 ) ] × [ ϕ T ( β + ½ α , t 2 ) - ϕ T ( β - ½ α , t 2 ) ] } .
D ( α ) = d f d f d t 1 d t 2 d h d h [ 1 - H ( f ) ] × [ 1 - H * ( f ) ] exp [ 2 π i ( f - f ) t ] exp [ - 2 π i ( f t 1 - f t 2 ) ] × { [ n ( β + ½ α + ϑ ˙ h t 1 , h ) - n ( β - ½ α + ϑ ˙ h t 1 , h ) ] × [ n ( β + ½ α + ϑ ˙ h t 2 , h ) - n ( β - ½ α + ϑ ˙ h t 2 , h ) ] } .
[ n ( β + ½ α + ϑ ˙ h t 1 , h ) - n ( β - ½ α + ϑ ˙ h t 1 , h ) ] × [ n ( β + ½ α + ϑ ˙ h t 2 , h ) - n ( β - ½ α + ϑ ˙ h t 2 , h ) ] = - ½ [ n ( β + ½ α + ϑ ˙ h t 1 , h ) - n ( β + ½ α + ϑ ˙ h t 2 , h ) ] 2 + ½ [ n ( β + ½ α + ϑ ˙ h t 1 , h ) - n ( β - ½ α + ϑ ˙ h t 2 , h ) ] 2 + ½ [ n ( β - ½ α + ϑ ˙ h t 1 , h ) - n ( β + ½ α + ϑ ˙ h t 2 , h ) ] 2 - ½ [ n ( β - ½ α + ϑ ˙ h t 1 , h ) - n ( β - ½ α + ϑ ˙ h t 2 , h ) ] 2 .
D ( α ) = d f d f d t 1 d t 2 d h d h × [ 1 - H ( f ) ] [ 1 - H * ( f ) ] × exp [ 2 π i ( f - f ) t ] exp [ - 2 π i ( f t 1 - f t 2 ) ] × C N 2 ( h + h 2 ) × { ½ [ α + ϑ ˙ ( h t 1 - h t 2 ) 2 + h - h 2 ] 1 / 3 + ½ [ α + ϑ ˙ ( h t 1 - h t 2 ) 2 + h - h 2 ] 1 / 3 - [ ϑ ˙ ( h t 1 - h t 2 ) 2 + h - h 2 ] 1 / 3 } .
h + = ½ ( h + h ) ,
h - = h - h ,
t + = ½ ( t 1 + t 2 ) ,
t - = t 1 - t 2 ,
f + = ½ ( f + f ) ,
f - = f - f .
D ( α ) = k 2 d f + d f - d t + d t - d h + d h - × [ 1 - H ( f + + ½ f - ) ] [ 1 - H * ( f + - ½ f - ) ] × exp ( 2 π i f - t ) exp [ - 2 π i ( f + t - + f - t + ) ] × C N 2 ( h + ) { ½ [ α + ϑ ˙ ( h - t + + h + t - ) 2 + h - 2 ] 1 / 3 + ½ [ α + ϑ ˙ ( h - t + + h + t - ) 2 + h - 2 ] 1 / 3 - [ ϑ ˙ ( h - t + + h + t - ) 2 + h - 2 ] 1 / 3 } .
ϑ ˙ h + t - ϑ ˙ h - t + ,
D ( α ) = k 2 d f + d f - d t + d t - d h + d h - × [ 1 - H ( f + + ½ f - ) ] [ 1 - H * ( f + - ½ f - ) ] × exp ( 2 π i f - t ) exp [ - 2 π i ( f + t - + f - t + ) ] × C N 2 ( h + ) { ½ [ α + ϑ ˙ h + t - ) 2 + h - 2 ] 1 / 3 + ½ [ α + ϑ ˙ h + t - ) 2 + h - 2 ] 1 / 3 - [ ϑ ˙ h + t - ) 2 + h - 2 ] 1 / 3 } .
D ( α ) = k 2 d f + d t - d h + d h - × [ 1 - H ( f + ) ] [ 1 - H * ( f + ) ] × exp ( - 2 π i t + t - ) C N 2 ( h + ) × { ½ [ α + ϑ ˙ h + t - 2 + h - 2 ] 1 / 3 + ½ [ α + ϑ ˙ h + t - 2 + h - 2 ] 1 / 3 - [ ϑ ˙ h + t - ) 2 + h - 2 ] 1 / 3 } .
d x [ ( A 2 + x 2 ) 1 / 3 - ( x 2 ) 1 / 3 ] = 2.91 A 5 / 3
D ( α ) = 2.91 k 2 d f d t d h 1 - H ( f ) 2 exp ( - 2 π i f t ) C N 2 ( h ) × [ ½ α + ϑ ˙ h t 5 / 3 + ½ α - ϑ ˙ h t 5 / 3 - ϑ ˙ h t 5 / 3 ] ,

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